You guys kind of are missing out on the time period that Woody's character embodies, a 1950s Saturday Morning cartoon of sorts. The Woody episode shown in TS2 seemed to play out a bit like some of the live action cowboy shows of the 50s that aired in the morning as opposed to prime time (at least the ones that had a family of sorts). So in shows like Sky King, while the protagonist there was a cowboy rancher who flew an airplane, I don't recall him ever pulling a gun on anybody. Roy Rogers on his 50s TV show was about the same way. When one got to cowboy shows in prime time, then the guns came out more (not sure when Lone Ranger was on though). Zorro opted out of using a gun and did his thing with a whip instead. The protagonists in the family shows typically never used guns as they wanted to show they were smarter than that. It also in a sense was competing with the Superman ethics on TV that were brought about by the 1950s Superman TV show as well (of course, Superman didn't need a gun, but he never tried to kill anyone either, instead usually just letting the villains do themselves in as they tried to attack him).
Plus, the "original" Woody was a marionette, sort of in throwback to Howdy Doody and Doody never had a gun either. By the 1960s, Gerry Anderson was working with such puppets and showed how VERY difficult it was to put a gun or any other prop in their hands. If Woody had a pistol, it would have really looked a bit awkward for the type of floppy puppet doll he was. They had plenty of guns with other characters as Buzz had his laser in the arm and the green army men had their plastic army rifles molded to them and both toys related to them had the guns in them for all to see. Heck, TS1 blew up a Combat Carl, which was apparently because GI Joe wasn't available from Hasbro for the nasty deed. Even with the name change, people knew exactly what it was and it had a gun as I can recall.
So you may ask why Woody has no gun. But to me, in the way he was portrayed, he didn't really need a gun and until it was mentioned, I don't recall even noticing he had no guns.